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Calc II - CU Boulder - Fall 2010 - Section005

Comparison Test

Tuesday, October 5, 2010 - 14:00 - Wednesday, October 6, 2010 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

9.4 Comparison Test Examples – October 5, 2010

  1. Use the comparison test to determine if the given series converges or diverges.

    \sum _{{n=1}}^{\infty}\frac{1}{n^{4}+1}

    Solution: Since


    and we know the p-series

    \sum _{{n=1}}^{\infty}\frac{1}{n^{4}}

    converges, the given series also converges.

Integral Test

Monday, October 4, 2010 - 14:00 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

9.3 Integral Test Examples – October 4, 2010

  1. What does Theorem 9.2 say about the convergence or divergence of the series?

    \sum _{{n=1}}^{\infty}\left(2+e^{{-n}}\right)

    Solution: Since the series terms are of the form 2+e^{{-n}}, and the limit of these terms as n\rightarrow\infty is 2 (not zero), we know that this series diverges.

  2. What does Theorem 9.2 say about the convergence or divergence of the series?

    \sum _{{n=1}}^{\infty}\sin n

Exam 2 Review 1

A review sheet for exam 2 is now available, as an attachment to this post.

LA Information

Wednesday, October 6, 2010 - 18:00 - 19:00

There is an informational session about becoming a Learning Assistant (LA) on Wednesday, October 6 at 6PM in UMC235. Applications for Spring 2011 will be available October 6 - October 20.

If you are interested in attending the informational session, e-mail olivia.holzman@colorado.edu

Geometric Series

Friday, October 1, 2010 - 14:00 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

9.2 Geometric Series Notes – October 1, 2010

A tip on recognizing geometric series: Many times when you're using convergence comparison tests, the series you use for comparison is either a p-series or a geometric series. In the first case, we know that

\sum _{{n=1}}^{\infty}\frac{1}{x^{p}}

converges for p>1 and diverges for p\le 1. In the second case, the geometric series in question may be more challenging to pick out. For instance, when you have

\sum _{{n=1}}^{\infty}\frac{1}{2^{n}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots

you may either view this as the geometric series

\sum _{{n=1}}^{\infty}\frac{1}{2}\left(\frac{1}{2}^{{n-1}}\right)

where a=1/2 and x=1/2, giving from the formulas for geometric series that

\sum _{{n=1}}^{\infty}\frac{1}{2^{n}}=\frac{1/2}{1-1/2}=\frac{1/2}{1/2}=1.

Or (and this may seem odd at first, but I think it yields faster computations) view this as the geometric series


where a=1 and x=1/2, except that the first term needs to be removed (since the series in question has no 1). So, we also have

\sum _{{n=1}}^{\infty}\frac{1}{2^{n}}=\frac{1}{1-1/2}-1=1.

Webwork 9.2 Hints

Math 2300 Section 005 – Calculus II – Fall 2010

Hints for Webwork Section 9.2

Most of the Webwork problems for this section simply use the notions of geometric series that we developed in class. There are a few, however, that are challenging and require you to put a lot of thought in to setting up the series.

5. Consider a similar example. Us math folk really like coffee. Assume that at the start of each day I drink 300mg of caffeine. Also assume that the “halflife” of this caffeine in my system is 7.5 hours. So, in 7.5 hours I will effectively have 150mg of caffeine in my system, and in 15 hours that will be halved again and I will now have 75mg of caffeine in my system. Notice that at 24 hours (a full day after my last sip of coffee) I still have some caffeine in my system. How much? The answer: I take my initial amount and multiply it by half for each period of 7.5 hours. And since 24/7.5 of those periods happen in 24 hours, we find that I have

300\cdot\frac{1}{2}^{{24/7.5}}\approx 32.64\,\text{mg}

of caffeine in my system after 24 hours. At that instant, I drink another 300mg of caffeine and I now have

Sequence and Series Quiz

Due Date: 
Friday, October 1, 2010 - 16:00 - Monday, October 4, 2010 - 16:00

This quiz is due on Monday at 4PM. Obviously, only questions 1 and 2 count for credit. Due to my current lack of creativity, there is no replacement question on this quiz.

Worksheet 5

The solutions are posted to worksheet #5


Wednesday, September 29, 2010 - 14:00 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

9.1 Sequences Examples – September 29, 2010

Fill in the empty cells of the following table.

\begin{array}[]{c|c}\textbf{General Term}&\textbf{First 5 Terms}\\<br />
\hline\hline&\\<br />
\hline&\\<br />
s_{n}=\frac{1}{n}&\\<br />
&\\<br />
\hline&\\<br />
s_{n}=n^{2}&\\<br />
&\\<br />
\hline&\\<br />
&1,2,4,8,16,\ldots\\<br />
&\\<br />
\hline&\\<br />
s_{m}=\frac{n}{n+1}&\\<br />
&\\<br />
\hline&\\<br />
&1,-1,1,-1,1,\ldots\\<br />
&\\<br />
\hline&\\<br />

We'll introduce the notions of convergent and divergent. You should be able to recognize convergent and divergent sequences by their graphs. We'll also introduce the Fibonacci sequence.

The solutions to the following examples are located below.

  1. Give the first five terms of the sequence


Fall 2010 Winning at Math Power Hours

I mentioned the Winning and Math Power Hours in class. I've attached a PDF info sheet to this post. Some of the topics that you may find particularly useful are the "success strategies, exam preparation and managing math anxiety" sessions.

© 2011 Jason B. Hill. All Rights Reserved.